GATE CS 2026 Set 2 Q.27: Calculus

For a real number $a$ , let $I(a) = \int_{-1}^{1} (3x^2 - ax + 1)\,dx$ . Which of the following statements is/are true?

The value of $I(a)$ is independent of the value of $a$

The value of $I(a)$ can vary with the value of $a$

There exists $a \in (-\infty, +\infty)$ such that $I(a)$ is a positive real number

There exists $a \in (-\infty, +\infty)$ such that $I(a)$ is a negative real number

Official answer and solution

Solution

Evaluate the definite integral over the symmetric interval $[-1, 1]$ using even and odd function properties.

Splitting the integrand into even and odd terms over $[-1, 1]$ :

I(a) = \int_{-1}^{1} (3x^2 + 1)\,dx - a \int_{-1}^{1} x\,dx

Since $x$ is an odd function, $\int_{-1}^{1} x\,dx = 0$ for every real $a$ . For the even part:

I(a) = 2 \int_{0}^{1} (3x^2 + 1)\,dx = 2 \left[x^3 + x\right]_0^1 = 2(1 + 1) = 4

Since $I(a) = 4 > 0$ for all $a \in \mathbb{R}$ , $I(a)$ is independent of $a$ (A) and is a positive real number for all $a$ (C).

Final answer

Options (A), (C)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Paper: GATE 2026 Computer Science and Information Technology - Session 2 (CS2)
Question: 27 of 65
Section: Computer Science and Information Technology

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